Establishing a precise lower bound is the new and delicate step. For all specific functions for which \(\text{{\rm HD}}(\mathcal {I}(f))\)  was known, such as elliptic and Nevanlinna functions mentioned above, the precise lower bound was obtained using McMullen's method in [1]}. However, for this method to give sharp estimates the poles must be very
uniformly distributed in addition with a uniform behavior of coefficients \(b_j\) , \(j\ge 1\) . This is clearly the case for elliptic and Nevanlinna functions as well
as for the examples in [2]}.
In order to treat our general case we had to proceed totally differently. Our idea was to use the concept of non-autonomous iterated function systems and some results proved for them. Indeed, in Section, we briefly summarize some facts from this theory. Then, in Section , we prove the inequality
\(\text{{\rm HD}}(\mathcal {I}(f))\ge \delta ,\) 
